Chern–simons Theory and the Asymptotic Expansion of Witten–reshetikhin–turaev’s Invariant of 3–manifolds

In this report we make a thorough study of the Chern–Simons field theory for compact oriented 3–manifolds associated to a compact simply connected simple Lie group G mainly following [F]. The action of the classical Wess–Zumino–Witten theory in 1 + 1 dimensions appears in Chern–Simons theory in the definition of the Hermitian line corresponding to the boundary of a 3–manifold. A part of the report is concerned with the study of the moduli space of flat G– connections on a compact oriented 3–manifold, the flat connections being the so-called classical solutions in the Chern–Simons field theory. We also give an outline of how to construct (in a rigorous way) the 2–dimensional part (the modular functor) of a 2 + 1–dimensional TQFT using geometric quantization of the moduli space of flat connections on Riemann surfaces. In the final part of the report, we begin a rigorous calculation of the large quantum level asymptotics of the SU(2) Reshetikhin–Turaev invariants of 3–fibered Seifert manifolds with base S following [Ro1]. The report supplements the calculations of Rozansky by obtaining analytic estimates needed to justify the method of Rozansky.